Let $X$ be a compact complex manifold with boundary and let $L^k$ be a high power of a hermitian holomorphic line bundle over $X.$ When $X$ has no boundary, Demailly’s holomorphic Morse inequalities give asymptotic bounds on the dimensions of the Dolbeault cohomology groups with values in $L^k,$ in terms of the curvature of $L.$ We extend Demailly’s inequalities to the case when $X$ has a boundary by adding a boundary term expressed as a certain average of the curvature of the line bundle and the Levi curvature of the...